608 ON THE PROBLEM OF RANDOM VIBRATIONS, AND [441
resultants, but for practical purposes it, requires transformation when we contemplate a very great n.
The necessary transformation can be obtained after Laplace with the aid of Stirling's theorem. The process is detailed in Todhunter's History of the Theory of Probability, p. 548, but the corrections to the principal term there exhibited (of the first order in x) do not appear here where the probabilities of the plus and 'minus alternatives are equal. On account of the symmetry, no odd powers of x can occur. I have calculated the resulting expression with retention of the terms which are of the order 1/w2 in comparison with the principal term. The resultant x itself may be considered to be of order not higher than \Jn.
By Stirling's theorem
(11)
where
with similar expressions for (^n — •£#)! and Qw. + £#)! For the moment we omit the correcting factors 0. Thus
L ~~ •
n*J {l+osjn For the logarithm of the product of the last two factors, we have
#"
5nB
__ -
~~2n 2n2 4m?
and for the product itself
^ jx + i (+ .
( 2w\?i The principal term in (10) is
There are still the factors (7 to be considered. We have
= Ji _L 1 | J1 _ 1 J_/l ~^i + I2n + 288n2} | 3n & "'^o-l ue of the term counted s onwards from the unique maximum is